Theorem 4.1 (Error bound for analytic function [1]}, [2]}) 

Let \(f\)  be a real function on \([-1, 1]\)  and \(f_M\)  be its \((M-1)\) -degree polynomial interpolation built on Chebyshev nodes \(\xi _k:=\cos {\frac{k - 1}{M - 1}\pi }\) , \(k=1,\dots ,M\) . If \(f\)  has an analytic extension in a Bernstein ellipse \(\mathcal {B}\)  with foci \(\pm 1\)  and major and minor semiaxis lengths summing up to \(\varrho > 1\)  such that \(\sup _{\mathcal {B}} |f| \le \frac{\varrho - 1}{4}\overline{C}\)  for some constant \(\overline{C}>0\) , then, for each \(M \ge 1\) , the following bound holds:
\(||f - f_M||_{L^{\infty }([-1, 1])}\le \overline{C}\varrho ^{1-M}.\) 

